Link to actual problem [2391] \[ \boxed {9 x^{2} y^{\prime \prime }+9 \left (-x^{2}+x \right ) y^{\prime }+y \left (x -1\right )=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Regular singular point. Difference not integer"}
type detected by Maple
[[_2nd_order, _with_linear_symmetries]]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\)\begin{align*} \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {{\mathrm e}^{\frac {x}{2}} \operatorname {WhittakerM}\left (\frac {11}{18}, \frac {1}{3}, x\right )}{\sqrt {x}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {x}\, {\mathrm e}^{-\frac {x}{2}} y}{\operatorname {WhittakerM}\left (\frac {11}{18}, \frac {1}{3}, x\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {{\mathrm e}^{\frac {x}{2}} \operatorname {WhittakerW}\left (\frac {11}{18}, \frac {1}{3}, x\right )}{\sqrt {x}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {x}\, {\mathrm e}^{-\frac {x}{2}} y}{\operatorname {WhittakerW}\left (\frac {11}{18}, \frac {1}{3}, x\right )}\right ] \\ \end{align*}